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The subspace is then called a retract of the original space. A deformation retraction is a mapping that captures the idea of continuously shrinking a space into a subspace. An absolute neighborhood retract (ANR) is a particularly well-behaved type of topological space. For example, every topological manifold is an ANR.
In mathematics, the Bing–Borsuk conjecture states that every -dimensional homogeneous absolute neighborhood retract space is a topological manifold. The conjecture has been proved for dimensions 1 and 2, and it is known that the 3-dimensional version of the conjecture implies the Poincaré conjecture.
edit] * The base point of a based space. X + {\displaystyle X_{+}} For an unbased space X, X + is the based space obtained by adjoining a disjoint base point. A absolute neighborhood retract abstract 1. Abstract homotopy theory Adams 1. John Frank Adams. 2. The Adams spectral sequence. 3. The Adams conjecture. 4. The Adams e -invariant. 5. The Adams operations. Alexander duality Alexander ...
"An absolute neighborhood retract (ANR) is a particularly well-behaved type of topological space. For example, every topological manifold is an ANR. Every ANR has the homotopy type of a very simple topological space, a CW complex."
Likewise, if is a closed subspace of and the subspace inclusion is an absolute neighborhood retract, then the inclusion of into is a cofibration. [ 2 ] [ 3 ] Hatcher's introductory textbook Algebraic Topology uses a technical notion of good pair which has the same long exact sequence in singular homology associated to a cofibration, but it is ...
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In the case that is a subspace of fulfilling the mild regularity condition that there exists a neighborhood of that has as a deformation retract, then the group ~ (/) is isomorphic to (,). We can immediately use this fact to compute the homology of a sphere.
Karol Borsuk (8 May 1905 – 24 January 1982) was a Polish mathematician.His main area of interest was topology.He made significant contributions to shape theory, a term which he coined.